The Gielis Super Shape Formula

Though the Danish poet and scientist Piet Hein (1905–1996) is
often credited with the discovery of the super-ellipse, it was the
geometer Gabriel Lamé (1818) who first generalized the concept
of an ellipse to a super-ellipse
by generalizing the exponents of the standard equation of an ellipse as
shown below:

This super-ellipse formula can be converted to the following
equivalent, polar coordinate form by substituting x = r.cos
θ and y = r.sin θ into the above equation
and solving for r to obtain:

More recently, the Belgian Botanist, Johan Gielis (1996, 2003) from
Antwerpen generalized the polar coordinate version further as follows
by considering possibly different values of the exponents and also
including a factor m/4 to divide the polar coordinate plane
into m sectors:

Note: In the dynamic JavaSketchpad sketch below, the
above formula is
represented. Drag the turquoise
pointsm, n1, n2,
n3, a or b to explore the huge and
intriguing variety of symmetric figures that are produced with the
Gielis super shape formula. (Unfortunately the slider values in the
JavaSketchpad sketch
change discreetly so only approximate values for whole numbers can
sometimes be obtained).

The Gielis Super Shape Formula

A fascinating aspect of the Gielis formula is that it can be
used to model the shape of numerous natural objects from plants and
flowers to snowflakes and crystals. For example, the shape shown above,
produced by the parameter values (6, 100, 38, 38, 0.5, 1) - in the
order (m, n1, n2,
n3, a, b) - resembles that of the
snowflake shown in the first picture below (and by making a = b,
we obtain a regular hexagon). Try the parameter values (6, 100, 100,
100, 1, 1) and you should get a shape similar to that of the second
snowflake below. The parameter values (4, 10, 19, 19, 0.9, 0.9) and (5,
2, 9, 9, 1, 1) produce shapes approximately corresponding to a
cross-section of a certain type of cactus and that of a starfish,
respectively shown by the 3rd and 4th figures below. (For the starfish
shape, you can resize the figure by dragging the red point on the x-axis.)

Use the above dynamic sketch to further explore the effect of the
parameters. Many other examples of shapes produced by the Gielis
formula can be found in the online encyclopedia, Wikipedia, at Super Formula.

The parameter m basically determines the number of
'sectors', 'bulges', 'hollows' or 'corners' of the shape. When m
= 0, the shape is a circle, but with m = 4, depending on the
value of the other parameters, we can get a square with four vertices,
or a shape like the cactus cross section with four 'corners' and four
'hollows', or a standard ellipse (which has two narrow 'bulges' and two
wide ones). If n1 > n2 = n3
the sides can tend to straight lines and with appropriate values,
regular polygons can be produced. The parameters a and b
determine the lengths of the major axes and therefore the size of the
shape. With higher even values of m, different values of a
and b, will produce alternate sides of different length.

The Gielis super shape formula can be generalized to 3D in at least
two different ways, for example, by rotating the 2D version around one
of the axis, or by using polar coordinates in 3D. The 2D formula itself
can obviously be further generalized by considering different
parameters for m in the two parts of the formula it occurs.
Though the creation of closed figures are important if one wants to
model closed shapes such as flowers or snowflakes, mathematically
nothing prevents us to modify the Gielis super shape formula by instead
of adding the two absolute values, we could simply subtract the second
one from the first to produce a Super Hyperbola formula as shown below:

In the dynamic JavaSketchpad sketch below, the
above Super Hyperbola formula is
represented. Have FUN dragging the turquoise
pointsm, n1, n2,
n3, a or b to explore the huge and
intriguing variety of symmetric and other figures that are produced with
this formula! Which set(s) of values for the parameters would produce regular hyperbola?

Super Hyperbola

Note: The slider values in the
JavaSketchpad sketch unfortunately change discreetly; so only approximate values for whole numbers can
mostly be obtained. Therefore, because of these approximations, each hyperbolic branch of the Java sketch for (5, 2, 2, 2, 1, 1) appears to 'split' into two as it goes off to infinity. Also be aware, that the JavaSketchpad sketch, for some values of the parameters, does not always correctly display asymptotes and branches going off to infinity. For more accurate explorations, it is better to use either Sketchpad or some other similar graphing software. Accurate Sketchpad 4 and 5 sketches for both the Gielis formula and the Super Hyperbola formula are available for downloading from Sketchpad Exchange at Gielis Super Shape Formula and Super Hyperbola.